Definite Integration of a concave upward function- Inequality

[mathisfun1]

View attachment 2917

 

[Dethrone]

On the last integral, is the upper bound "9"?

 

[mathisfun1]

On the last integral, is the upper bound "9"?

Yes.

 

[MarkFL]

Hello and welcome to MHB! :D

Can you tell us what you've tried or what your thoughts are on how to begin, so our helpers have an idea where you are stuck and then can provide better help?

 

[mathisfun1]

Hello and welcome to MHB! :D

Can you tell us what you've tried or what your thoughts are on how to begin, so our helpers have an idea where you are stuck and then can provide better help?

Thank you so much!
I began with the condition that the secant must lie above the curve at all points if the curve is concave upward but that seem to lead nowhere. I also tried to utilise the fact that the double derivative of f(X) will be positive but to no avail.
Any help will be appreciated.

 

[MarkFL]

I think I would begin with the minimal case:

\(\displaystyle 0<f^{\prime\prime}(x)\)

So, integrating, what can you say about $f$?

 

[mathisfun1]

I think I would begin with the minimal case:

\(\displaystyle 0<f^{\prime\prime}(x)\)

So, integrating, what can you say about $f$?

Wouldn't integrating this inequality conclude the same thing as what we are given, that f is a concave upward function?

 

[MarkFL]

Wouldn't integrating this inequality conclude the same thing as what we are given, that f is a concave upward function?

What did you get when integrating?

 

[mathisfun1]

What did you get when integrating?

That f(x) is greater than zero, but that seems to be contradictory as a negative function may also have positive derivative. :(

 

[MarkFL]

Let's begin with:

\(\displaystyle 0<f^{\prime\prime}(x)\)

And so, on \(\displaystyle [1,k]\) where \(\displaystyle 1<k\in\mathbb{R}\) there must be some constant $C$ such that:

\(\displaystyle C<f^{\prime}(x)\)

What happens if you integrate again?

 

[mathisfun1]

Let's begin with:

\(\displaystyle 0<f^{\prime\prime}(x)\)

And so, on \(\displaystyle [1,k]\) where \(\displaystyle 1<k\in\mathbb{R}\) there must be some constant $C$ such that:

\(\displaystyle C<f^{\prime}(x)\)

What happens if you integrate again?

If f1(X)>f2(x)
doesn't implies that the indefinite integration of f1(x) will be greater than f2(x). This is only applicable when both sides are integrated within limits and thus when integrating from 0 to x, constant gets eliminated.

 

[MarkFL]

Let's try another approach...you mentioned that if the function is concave up, then it must lie beneath its secant line, or:

\(\displaystyle f(x)<\frac{f(b)-f(a)}{b-a}(x-a)+f(a)\)

What do you find upon integrating both sides from $a$ to $b$?